complex numbers

Front

What is the definition of a complex number?

Back

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, which means i^2 = -1.

Front

How do you simplify powers of 'i'?

Back

To simplify powers of 'i', use the pattern: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, and then repeat every four powers.

Front

What is the result of i^66?

Back

i^66 simplifies to -1.

Front

How do you multiply two complex numbers?

Back

To multiply two complex numbers (a + bi) and (c + di), use the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi^2, then simplify using i^2 = -1.

Front

What is the product of (1-3i) and (2+5i)?

Back

The product (1-3i)(2+5i) simplifies to 17 - i.

Front

How do you perform addition of complex numbers?

Back

To add complex numbers, combine the real parts and the imaginary parts separately: (a + bi) + (c + di) = (a + c) + (b + d)i.

Front

What is the result of (4 + 7i) - (3 + 2i)?

Back

The result of (4 + 7i) - (3 + 2i) simplifies to 1 + 5i.

Front

How do you find the square root of a negative number?

Back

To find the square root of a negative number, express it in terms of 'i': √(-x) = i√x.

Front

What is the square root of -36?

Back

The square root of -36 is 6i.